Theorem isstruct2 17081

Description: The property of being a structure with components in (1^st ‘𝑋)...(2^nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
isstruct2	⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Proof of Theorem isstruct2

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brstruct 17080	. . 3 ⊢ Rel Struct
2	1	brrelex12i 5680	. 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3		ssun1 4131	. . . . 5 ⊢ 𝐹 ⊆ (𝐹 ∪ {∅})
4		undif1 4429	. . . . 5 ⊢ ((𝐹 ∖ {∅}) ∪ {∅}) = (𝐹 ∪ {∅})
5	3, 4	sseqtrri 3984	. . . 4 ⊢ 𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅})
6		simp2 1138	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → Fun (𝐹 ∖ {∅}))
7	6	funfnd 6524	. . . . . 6 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}))
8		elinel2 4155	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ (ℕ × ℕ))
9		1st2nd2 7975	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℕ × ℕ) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
11	10	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
12	11	fveq2d 6839	. . . . . . . 8 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) = (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
13		df-ov 7364	. . . . . . . . 9 ⊢ ((1st ‘𝑋)...(2nd ‘𝑋)) = (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
14		fzfi 13900	. . . . . . . . 9 ⊢ ((1st ‘𝑋)...(2nd ‘𝑋)) ∈ Fin
15	13, 14	eqeltrri 2834	. . . . . . . 8 ⊢ (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩) ∈ Fin
16	12, 15	eqeltrdi 2845	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) ∈ Fin)
17		difss 4089	. . . . . . . . 9 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹
18		dmss 5852	. . . . . . . . 9 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
20		simp3 1139	. . . . . . . 8 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom 𝐹 ⊆ (...‘𝑋))
21	19, 20	sstrid 3946	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ⊆ (...‘𝑋))
22	16, 21	ssfid 9174	. . . . . 6 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ∈ Fin)
23		fnfi 9107	. . . . . 6 ⊢ (((𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}) ∧ dom (𝐹 ∖ {∅}) ∈ Fin) → (𝐹 ∖ {∅}) ∈ Fin)
24	7, 22, 23	syl2anc 585	. . . . 5 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) ∈ Fin)
25		p0ex 5330	. . . . 5 ⊢ {∅} ∈ V
26		unexg 7691	. . . . 5 ⊢ (((𝐹 ∖ {∅}) ∈ Fin ∧ {∅} ∈ V) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
27	24, 25, 26	sylancl 587	. . . 4 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
28		ssexg 5269	. . . 4 ⊢ ((𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅}) ∧ ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐹 ∈ V)
29	5, 27, 28	sylancr 588	. . 3 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝐹 ∈ V)
30		elex 3462	. . . 4 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ V)
31	30	3ad2ant1 1134	. . 3 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 ∈ V)
32	29, 31	jca 511	. 2 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∈ V ∧ 𝑋 ∈ V))
33		simpr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
34	33	eleq1d 2822	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
35		simpl 482	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
36	35	difeq1d 4078	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
37	36	funeqd 6515	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
38	35	dmeqd 5855	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
39	33	fveq2d 6839	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
40	38, 39	sseq12d 3968	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
41	34, 37, 40	3anbi123d 1439	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
42		df-struct 17079	. . 3 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
43	41, 42	brabga 5483	. 2 ⊢ ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
44	2, 32, 43	pm5.21nii 378	1 ⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  {csn 4581  ⟨cop 4587   class class class wbr 5099   × cxp 5623  dom cdm 5625  Fun wfun 6487   Fn wfn 6488  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  Fincfn 8888   ≤ cle 11172  ℕcn 12150  ...cfz 13428   Struct cstr 17078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12151  df-n0 12407  df-z 12494  df-uz 12757  df-fz 13429  df-struct 17079

This theorem is referenced by:  structn0fun  17083  isstruct  17084  setsstruct2  17106